## 412 Folded Plates

FIGURE 4.26 Folded plate roofs. (a) Solid plates. (b) Trussed plates.

FIGURE 4.26 Folded plate roofs. (a) Solid plates. (b) Trussed plates.

The plates may be constructed in different ways. For example, each plate may be a stiffened steel sheet or hollow roof decking (Fig. 4.26a). Or it may be a plate girder with solid web. Or it may be a truss with sheet or roof decking to distribute loads transversely to the chords (Fig. 4.26b).

A folded-plate structure has a two-way action in transmitting loads to its supports. In the transverse direction, the plates act as slabs spanning between plates on either side. Each plate then serves as a girder in carrying the load received from the slabs longitudinally to the supports.

The method of analysis to be presented assumes the following: The material is elastic, isotropic, and homogeneous. Plates are simply supported but continuously connected to adjoining plates at fold lines. The longitudinal distribution of all loads on all plates is the same. The plates carry loads transversely only by bending normal to their planes and longitudinally only by bending within their planes. Longitudinal stresses vary linearly over the depth of each plate. Buckling is prevented by adjoining plates. Supporting members such as diaphragms, frames, and beams are infinitely stiff in their own planes and completely flexible normal to their planes. Plates have no torsional stiffness normal to their own planes. Displacements due to forces other than bending moments are negligible.

With these assumptions, the stresses in a steel folded-plate structure can be determined by developing and solving a set of simultaneous linear equations based on equilibrium conditions and compatibility at fold lines. The following method of analysis, however, eliminates the need for such equations.

Figure 4.27a shows a transverse section through part of a folded-plate structure. An interior element, plate 2, transmits the vertical loading on it to joints 1 and 2. Usual procedure is to design a 1-ft-wide strip of plate 2 at midspan to resist the transverse bending moment. (For continuous plates and cantilevers, a 1-ft-wide strip at supports also would be treated in the same way as the midspan strip.) If the load is w2 kips per ft2 on plate 2, the maximum

FIGURE 4.27 Forces on folded plates. (a) Transverse section. (b) Forces at joints 1 and 2. (c) Plate 2 acting as girder. (d ) Shears on plate 2.

bending moment in the transverse strip is w2h2a2/8, where h2 is the depth (feet) of the plate and a2 is the horizontal projection of h2.

The 1-ft-wide transverse strip also must be capable of resisting the maximum shear w2h2/2 at joints 1 and 2. In addition, vertical reactions equal to the shear must be provided at the fold lines. Similarly, plate 1 applies a vertical reaction W1 kips per ft at joint 1, and plate 3, a vertical reaction w3h3/2 at joint 2. Thus the total vertical force from the 1-ft-wide strip at joint 2 is

Similar transverse strips also load the fold line. It may be considered subject to a uniformly distributed load R2 kips per ft. The inclined plates 2 and 3 then carry this load in the longitudinal direction to the supports (Fig. 4.27c). Thus each plate is subjected to bending in its inclined plane.

The load to be carried by plate 2 in its plane is determined by resolving R1 at joint 1 and R2 at joint 2 into components parallel to the plates at each fold line (Fig. 4.27b). In the general case, the load (positive downward) of the nth plate is pn = u Rn + -IT^n (4.152)

kn cos fan kn-1 cos fan where Rn = vertical load, kips per ft, on joint at top of plate n

Rn-1 = vertical load, kips per ft, on joint at bottom of plate n fan = angle, deg, plate n makes with the horizontal kn = tan fan - tan fan+1

This formula, however, cannot be used directly for plate 2 in Fig. 4.27(a) because plate 1 is vertical. Hence the vertical load at joint 1 is carried only by plate 1. So plate 2 must carry

k2 cos fa2

To avoid the use of simultaneous equations for determining the bending stresses in plate 2 in the longitudinal direction, assume temporarily that the plate is disconnected from plates 1 and 3. In this case, maximum bending moment, at midspan, is

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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